Problem solving is one of the critical and central activities in one’s life.

Problems come in all shapes, sizes, varieties, and levels of difficulty.

Problems grow more complex each year.

Problem solving can be easier, more effective, and more fun if you have a flexible system for solving problems.

There is no substitute for experience. If you want to become a better problem solver, you must practice, practice, practice. Hence, the more problem solving you do, the better problem solver you become.

Some tools for solving problems…..

PROBLEM SOLVING:

A Student’s Guide

Rule 1

If at all possible, avoid reading the problem. Reading the problem only consumes time and causes confusion.

Rule 2

Extract the numbers from the problem in the order in which they appear. Pay no attention for numbers written in words.

Rule 3

If rule 2 yields three or more numbers, the best bet for getting the answer is adding them together.

Rule 4

If there are only two numbers which are approximately the same size, then subtraction should give the best results.

Rule 5

If there are only two numbers in the problem and one is much smaller than the other, then divide if it goes evenly-otherwise, multiply.

Rule 6

If the problem seems like it calls for a formula, pick a formula that has enough letters to use all the numbers given in the problem.

Rule 7

Never, never spend too much time solving problems.

This set of rules will get you through even the longest assignment in the minimum time with little or no thinking.

Tools That May Really Help

Problem Solving Tools You May Use

1. Rephrasing:

Often a problem seems complex or hard to understand simply because the words used are complicated, vague, or confusing. By rephrasing the problem in your own words, you can get it organized in your mind. Put the problem in your own words until you feel comfortable with your understanding of the problem.

Try stating the goal in your own words and as completely as you possibly can.

2. Possibility listing:

One of the easiest and most effective ways to get control of a confused situation is simply to itemize the variables and possibilities involved. This involves making a list of the key factors involved. In this case the further analysis of the puzzle can be transformed into a list of factors that make the puzzle a problem.

Try listing the variables and factors of the problem.

3. Identify sub goals:

When a problem is complex, breaking it down into sub problems and solving each part is helpful. By analyzing the problem carefully and not being distracted by the first thing that comes to mind, you may be able to discover the one key factor that lies at the heart.

Try simplifying the problem or the puzzle by breaking it down into sub-problems and then solving the parts.

4. Trial and error:

This is the weakest and often the most inefficient method. It is randomly trying one possibility, then another, and then another. This method is also called guess and check. The correct solution is discovered by chance. This method is testing all the possibilities at random. (It is very probable you will use other methods instead of making a completely exhaustive search)

Try guessing and checking your solution

5. Estimate, predict or project

Get an idea what the solution would be close to. Predict the range of where the answer might be.

Try estimating what the answer would be close to

6. Best first analysis:

This searching strategy involves testing the most probable or most desirable (or promising) possibility first. This method can also be used on sub goals. If the first method attempted fails to produce a solution the second best choice is tried.

Try the most desirable choice first.

7. Worst first analysis:

This searching strategy involves testing the least probable or desirable (or promising) possibility first. This method can also be used on sub goals. If the first method attempted fails to produce a solution the second least desirable choice is tried.

Try the least desirable choice first.

8. Process of Elimination:

This method is organizing the possibilities by eliminating what does not work. This process may be used to solve sub goals and categorizing trial and error testing.

Try eliminating the possibilities that do not work.

9. Jump the Track:

Often problem solvers get stuck in a mental rut and do the same process over and over. Stopping to reconsider the whole course of your attack on the problem may help. Start again with a completely different approach or a different point of view. Enlarge the range of options to include unusual ones.

Try a totally different approach.

10. Look for patterns:

By examining the puzzle carefully, a pattern for arranging the pieces or in the solution may be observed. This may be patterns in shapes, color, size, process of steps or a hidden code.

Try looking for a hidden pattern.

11. Draw or use a diagram, table, or model:

Problems are often approached by sketching out the process on paper. Often Athinking with a pencil@ helps clarify the thinking process.

Try looking using a pencil to sketch or keep track of your thinking process.

12. Work backwards:

When the goal is clear, you can begin there and work backwards. Taking a completed puzzle apart piece by piece, or working a maze backwards or completing describing the finished puzzle may help in the process.

Try working backwards by understanding what the solved problem must look like.

13. Simplify

Do a simpler problem of the same kind to understand the method. Apply that method to the present problem.

Try doing a simpler problem of the same kind and apply that method.

14. Logic

When there are steps that depend on each other, decide which step goes first. After that, decide the steps that follow in a reasonable order. Discover how the steps fit together with phrases such as: If I do this, then this will happen.

Try breaking the steps of problem into a reasonable order.

15. Act it Out

Often it helps to play act the problem by demonstrating the situation physically.

Try play acting the problem by demonstrating the situation.

16. Create an equation

Practice some algebra by using letters as variables to represent unknown quantities. Solving the equation leads to the solution of the problem

Often teachers think that gifted students are those who just need more intellectual stimulation. The phase “they can take care of themselves” is often heard. There are other needs that must be addressed to serve the needs of the gifted students. Here are just a few…………….

Perfectionism: The ability to see how one might ideally perform, combined with emotional intensity leads many gifted children to unrealistically high expectations of themselves.

Underachievement: This is the discrepancy between potential and performance or ability and achievement. When a gifted student is not working up to his or her potential this is called underachievement.

Avoidance of risk taking: In the same way the gifted see the possibilities, they also see potential problems in undertaking those activities. Avoidance of potential problems can mean avoidance of risk-taking, and may result in underachievement.

Uneven development: Motor skills, especially fine-motor, often lag behind cognitive conceptual abilities. These children may see in their “minds eye” what they want to do, construct, or draw. However, motor skills do not allow them to achieve the goal. Intense frustration and emotional outbursts may result.

Multi-potentiality: Gifted children often have advance capabilities and may be involved in diverse activities to an almost frantic degree. Though seldom a problem for the child, this may create problems for the family, as well as quandaries when decisions must be about career selection.

Peer relationships: Gifted students find that they often need both social and intellectual peers and they need to develop relationships with both.

Excessive self-criticism: The ability to see possibilities and alternatives may imply that youngsters see idealistic images of what they might be, and simultaneously berate themselves because they see how they are falling short of an ideal.

Emotional intensity and stress: Because of the areas stated above and the uneven coping abilities, gifted students may feel deeper and may experience intense stress.

Social Skills: Often gifted students put much emphasis on the advanced thought process to the neglect of social skills that seem to come naturally to others. These areas could be listening skills, communication skills, and friendship skills.

As a well-spent day brings happy sleep, so life well used brings happy death.

Iron rusts from disuse; water loses its purity from stagnation and in cold weather becomes frozen; even so does inaction sap the vigor of the mind.

Patience serves as a protection against wrongs as clothes do against cold. For if you put on more clothes as the cold increases, it will have no power to hurt you. So in like manner you must grow in patience when you meet with great wrongs, and they will then be powerless to vex your mind.

You do ill if you praise, but worse if you criticize, what you do not understand.

Simplicity is the ultimate sophistication.

Obstacles cannot crush me. Every obstacle yields to stern resolve. He who is fixed to a star does not change his mind.

Every now and then go away, have a little relaxation, for when you come back to your work your judgment will be surer since to remain constantly at work will cause you to lose power of judgment. Go some distance away because then the work appears smaller, and more of it can be taken in at a glance, and a lack of harmony or portion is more readily seen.

For once you have tasted flight you will walk the earth with your eyes turned skywards, for there you have been and there you will long to return.

He who loves practice without theory is like the sailor who boards ship without a rudder and compass and never knows where he may cast.

I have been impressed with the urgency of doing. Knowing is not enough; we must apply. Being willing is not enough; we must do.

I have offended God and mankind because my work didn’t reach the quality it should have.

I love those who can smile in trouble, who can gather strength from distress, and grow brave by reflection. ‘Tis the business of little minds to shrink, but they whose heart is firm, and whose conscience approves their conduct, will pursue their principles unto death.

It had long since come to my attention that people of accomplishment rarely sat back and let things happen to them. They went out and happened to things.

Life is pretty simple: You do some stuff. Most fails. Some works. You do more of what works. If it works big, others quickly copy it. Then you do something else. The trick is the doing something else.

The noblest pleasure is the joy of understanding.

Where the spirit does not work with the hand, there is no art.

Who sows virtue reaps honor.

Anyone who conducts an argument by appealing to authority is not using his intelligence; he is just using his memory.

Study the science of art and the art of science.

Learn how to see and remember that everything is connected to everything else.

Good students naturally thirst after knowledge.

There are three classes of people. Those who see: those who see when they are shown: those who do not see.

POCATELLO, IDAHO – A new spin has been put on mathematics as Tendoy Elementary students use some magic to study various math concepts.

Bob Bishop, the Math Magician, has delighted students in kindergarten through sixth grade and teachers with his magic skills and math abilities over the past week.

“Math is so necessary in life,” he said. “It’s not just making math fun, but it’s also trying to attach some sense of understanding for students.”

Fifth grade teacher Vicki Reeder’s class had the opportunity to spend some time with Bishop while working on problem solving skills.

Students worked with calculators, the box of magic, learned how to do multiplication tables with their fingers, played a game called fast and loose and other activities.

During a game of fast and loose, Bishop produced a single chain and proceeded to fold it into a series of loops.

Students were asked to pick a loop and place their finger inside it. If they had guessed correctly the loop would stay around their finger. However, if they guessed incorrectly, the loop would slip away.

“You will win if you know mathematics, but you’ll lose if you don’t,” Bishop said.

Students learned how to follow the loops and determine the correct place to put their fingers.

Bishop has been performing for students and other audiences for 10 years and says he continually teaches students and teachers how math can be fun.

He said many students work with arithmetic but don’t fully understand problem solving skills.

With the help of a little magic, students are forced to observe the environment around them for any changes and think about possible outcomes.

“Generally students don’t really care to do math because it’s not fun,” Bishop said. “By making it interesting and proving to them they can do it, it helps to raise their self-esteem and interest level in math.”

Bishop will perform along with Tendoy Elementary students at 6:30 p.m. today for a Math Night.

Fifth grade student Quinci Shelley is acting as Bishop’s assistant during the show and said she can’t wait to perform for other students.

“I think it’s cool and it’s a good opportunity for us,” she said. “Some people don’t like math, but when they see this show it sparks their interest.”

Fifth grade student Brant Leo will lead the audience in applause, but said working with Bishop has been great because he’s learned new things.

“He’s helping students to improve their math by using cool tricks,” he said.

Bishop also worked with teachers after school and gave them various activities they can do with students in their classrooms.

“By making math fun, students will learn to enjoy it more and it will give them a sense of pride as they figure out difficult problems,” he said.

The issue of the definition of what a game is has open up many opinions. It has been said that the simplest questions are the most difficult. I would like to apply the lessons of strategy games to teaching.

Is there enough agreement of the definition of the word ‘game’ so it can be used as an adequate metaphor for life or at least some aspects of life? I believe every game has some sort of strategy. Given that every player suspends disbelief and enters the spirit of the game, every player has a method in which they use to seek to win the game. Can we assume that this is true with life? Would it be too much to say that every person has a strategy for life whether they have articulated or not? Perhaps it is easier to confine this idea to a particular task or assignment. What is the method or strategy that a person uses to accomplish a puzzle?

I do this often with my students. As I give them an assignment or a problem I walk around the room and ask them, “What is your method? What is your strategy?”

What I mean to do is for the student to be aware of his thinking method. I am asking the student to practice metacognition which for many is very difficult. When asked, “How did you arrive at that conclusion many students would say, ‘I don’t know I just did’”.

Arthur L. Costa says, “We can determine if students are becoming more aware of their own thinking if they are able to describe what goes on in their head when they think. When asked, they can describe what they know and what they need to know. They can describe their plan of action before they begin to solve a problem; they can list the steps and tell where they are in the sequence of a problem solving strategy; they can trace the pathways and blind alleys they took on the road to a problem solution.

They can apply cognitive vocabulary correctly as they describe their thinking skills and strategies. We will hear students using such terms and phrases as: “I have an hypothesis…,” “My theory is…,” “When I compare these points of view…,” “By way of summary…,” “What I need to know is…,” or “The assumptions on which I am working are…”

As an experiment start asking students what their strategy is for simple tasks and ask them the same question for more difficult tasks. Hopefully as they become used to this and learn to articulate their mental process they can begin to see similar strategies with more complex tasks.

I started today by teaching my students some basic “row” games based on Tic Tac Toe. We talked about how well known Tic Tac Toe was and transferred this knowledge to more complex games. We discussed how intuitive the rules of these other games were because they had a connection to this simple game. This laid the groundwork for the principles of learning by drawing on past knowledge and applying it to new situations

Some of those games were:

Abstract Strategy Game Checklist

Nim

Dots and Boxes

Dodgum

L Game

3 Spot Game

Hex

Cathedral

Tetra Trax

Isolation

Goblet

Quivive

Lotus

Quixo

Othello

Abalone

Pylos

Quoridor

Quarto

Quits

Mancala

Penta

Input

Score Four

Twixt

Qubic

Stadium Checkers

Stay Alive

Connect Four

Rubik‘s Magic Strategy Game

Slide Five

Bolix

Zertz

Paradux

Shift Tac Toe

Today’s goal was to learn how to play three games and to sense the learning process from learning the rules, playing a practice game where they learn to observe, and then to some basic strategy. When I played one boy a game of Bolix I lead him to a double two way win to demonstrate the depth of a simple (elegant) game. His response was, “My head hurts”. In my chess club a similar occurrence happened when the younger students murmured, “This is too hard”. Perhaps Samuel Goldwyn said it well, “If I look confused it’s because I’m thinking.”
Knowing that my pedagogy may be some of the issue, I do recognize that many students do not understand how to learn. This brings me to the quote…..

“Thinking is what you do when you do not know the answer”

Intelligent behavior is performed in response to questions and problems in which the answers are NOT immediately known.

This is one reason I teach strategy. How a person plays a game reflects how they think in other areas. Plato once said, “You can discover more about a person in an hour of play than in a year of conversation.